Not only could exist before the bible but did exist before the bible. And when I say
did exist I mean it in a way that the religion of said god existed (calm down militant athiests). There are many Christian faiths that predated the bible: Roman Catholicism, Eastern Orthodoxy, Coptic Christianiany, Syrian Christianity, Indian Christianity; There are many that I can label here.
Really the only Christianity to exist after the bible is the Protestant denominations that we know of today, such as The Baptists, the Anabaptists (Amish, Menonites, Hutterites), the Lutherans, the Calvanists, the Methodists, the Anglicans, The Waldenses, The Presbyterians, the Pentacostals, the Purtians, this list goes on and on and new groups are made up every year.
It is really the difference between the before the bible christians and the post-bible christians.
For some the "faith" would have endured lack of wrtiting or reading, and could have existed as an oral tradition. For others they need a book to believe in.